Micromachined Resonating Beam Gyroscopes

ABSTRACT

A single-axis resonating beam gyroscope uses a special arrangement of support tethers that maximizes the Q (quality factor) and minimizes stress sensitivity. The tethers are located at the nodal points of the beam with respect to a predetermined drive mode and are approximately one-fourth the length of the beam. Also, the tethers do not extend above or through the nodal points of the beam, which would be difficult to produce in typical MEMS fabrication processes. Embodiments typically use electrostatic drive and sense transduction. Trim electrodes may be used to compensate for any erroneous modal coupling.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This patent application claims the benefit of U.S. Provisional Patent Application No. 62/169,547 entitled MICROMACHINED GYROSCOPES filed on Jun. 1, 2015, which is hereby incorporated herein by reference in its entirety.

The subject matter of this patent application may be related to the subject matter of commonly-owned U.S. patent application Ser. No. ______ entitled MICROMACHINED CROSS-HATCH VIBRATORY GYROSCOPES filed on even date herewith, which is hereby incorporated herein by reference in its entirety.

FIELD OF THE INVENTION

The present invention relates generally to micromachined gyroscopes.

BACKGROUND OF THE INVENTION

Micromachined (MEMS) gyroscopes have become established as useful commercial items. Generally speaking, a MEMS gyroscope incorporates two high-performing MEMS devices, specifically a self-tuned resonator in the drive axis and a micro-acceleration sensor in the sensing axis. Gyroscope performance is very sensitive to such things as manufacturing variations, errors in packaging, driving, linear acceleration, and temperature, among other things. Basic principles of operation of angular-rate sensing gyroscopes are well understood and described in the prior art.

The principles of vibratory sensing angular rate gyroscopes with discrete masses are long-established. Generally speaking, a vibratory rate gyroscope works by oscillating a proof mass (also referred to herein as a “shuttle” or “resonator”). The oscillation is generated with a periodic force applied to a spring-mass-damper system at the resonant frequency. Operating at resonance allows the oscillation amplitude to be large relative to the force applied. When the gyroscope is rotated, Coriolis acceleration is generated on the oscillating proof mass in a direction orthogonal to both the driven oscillation and the rotation. The magnitude of Coriolis acceleration is proportional to both the velocity of the oscillating proof mass and the rotation rate. The resulting Coriolis acceleration can be measured by sensing the deflections of the proof mass. The electrical and mechanical structures used to sense such deflections of the proof mass are referred to generally as the accelerometer.

SUMMARY OF EXEMPLARY EMBODIMENTS

In certain embodiments there is provided a vibrating beam gyroscope comprising a rectangular beam arranged along a longitudinal axis in a device plane, the rectangular beam having a plurality of nodal points on both sides of the beam in the device plane with respect to a fundamental or higher order flexural drive mode, said nodal points being remote from the ends of said beam such that said beam includes a tail portion at each end of said beam; a set of tethers in the device plane, each tether coupled to the beam at a distinct nodal point, wherein a top edge of each tether is at or below a top edge of the beam; a set of drive electrodes configured for driving the beam to resonate in the drive mode; and a set of sense electrodes configured for sensing deflections of the beam caused by rotation of the gyroscope about the longitudinal axis.

In certain other embodiments there is provided a method of operating a resonating beam gyroscope, comprising driving for driving a rectangular beam to resonate in a fundamental or higher order flexural drive mode, the beam arranged along a longitudinal axis in a device plane, the beam supported by a set of tethers coupled to the beam at nodal points on both sides of the beam in the device plane with respect to the fundamental or higher order flexural drive mode, said nodal points being remote from the ends of said beam such that said beam includes a tail portion at each end of said beam, wherein a top edge of each tether is at or below a top edge of the beam; and sensing deflections of the beam caused by rotation of the gyroscope about the longitudinal axis.

In certain other embodiments there is provided a gyroscope comprising a resonator means including a rectangular beam arranged along a longitudinal axis in a device plane and a set of tethers coupled to the beam at nodal points on both sides of the beam in the device plane with respect to a fundamental or higher order flexural drive mode, said nodal points being remote from the ends of said beam such that said beam includes a tail portion at each end of said beam, wherein a top edge of each tether is at or below a top edge of the beam; means for driving the beam to resonate in the drive mode; and means for sensing deflections of the beam caused by rotation of the gyroscope about the longitudinal axis. In various alternative embodiments, the drive mode may be out-of-plane and the deflections may be in-plane, or the drive mode may be in-plane and the deflections may be out-of-plane. The relative length of the tail portions may be characterized by a node ratio of the beam Lnode/L approximately equal to X/(2*(X+1)), where Lnode is the length from the center of the beam to a nodal point, L is the total length of the beam, and X is the order of the flexural drive mode for the beam. Devices may be fabricated such that the tethers are configured to maximize a quality factor of the beam. The electrodes may be electrostatically coupled with the beam.

In any of the above-described embodiments, modal coupling may be mitigated by providing compensation signals to sets of variable-overlap trim electrodes to produce forces that reduce erroneous beam deflections. For example, the sets of variable-overlap trim electrodes may include a first set of trim electrodes including at least one electrode placed along a center portion on one side of the beam and at least one electrode placed along each tail section on the other side of the beam, and a second set of trim electrodes opposing the first set of trim electrodes, wherein the trim electrodes variably overlap the beam with respect to a direction of such deflections and are configured to produce forces in the direction of such deflections to compensate for erroneous beam deflections in the direction of such deflections. The erroneous beam deflections may be in proportion to resonation of the beam in the drive mode.

Additional embodiments may be disclosed and claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing features of embodiments will be more readily understood by reference to the following detailed description, taken with reference to the accompanying drawings, in which:

FIG. 1 is a schematic diagram showing a top view of the resonator and support tethers for a single-axis resonating beam gyroscope, in accordance with certain exemplary embodiments;

FIG. 2 is a schematic diagram showing conceptual components of a resonant beam of the resonator shown in FIG. 1, in accordance with exemplary embodiments of the invention;

FIG. 3 is a schematic diagram depicting two parameters for determining the node ratio of a resonant beam, in accordance with one exemplary embodiment;

FIG. 4 schematically shows a first possible drive mode and corresponding sense mode for a single-axis gyroscope of the type shown in FIG. 1, in accordance with one exemplary embodiment;

FIG. 5 schematically shows one possible arrangement of drive and sense electrodes to support the mode shapes shown in FIG. 4, in accordance with one exemplary embodiment;

FIG. 6 is a schematic diagram for an exemplary gyroscope drive and sense circuit for the gyroscope arrangement shown in FIG. 5, in accordance with one exemplary embodiment;

FIG. 7 schematically shows a set of optional electrodes placed adjacent to the tail ends of the beam for driving, sensing, or adjusting motion of the beam, in accordance with one exemplary embodiment; and

FIG. 8 is a schematic diagram showing an arrangement of split electrodes for reducing modal coupling, in accordance with one exemplary embodiment.

It should be noted that the foregoing figures and the elements depicted therein are not necessarily drawn to consistent scale or to any scale. Unless the context otherwise suggests, like elements are indicated by like numerals.

DETAILED DESCRIPTION OF SPECIFIC EMBODIMENTS

In embodiments of the present invention, a single-axis resonating beam gyroscope uses a special arrangement of support tethers that maximizes the Q (quality factor) while limiting stress sensitivity.

For purposes of the following description and the accompanying claims, a “set” includes one or more members, the “mode” of a resonating body is the shape of motion of the body at resonance, the term “anti-phase” with respect to the resonant modes (i.e., displacement) of two resonating bodies means that the resonating bodies resonate with the same mode shape but 180 degrees out-of-phase, the term “in-plane” with respect to a resonant mode means resonance predominately in the plane of the resonator structure(s), the term “out-of-plane” with respect to a resonant mode means resonance predominately normal to the plane of the resonator structure(s), a “node” or “nodal point” with respect to a resonating body is a point or area of the resonant motion having zero or near zero displacement, an “anti-node” with respect to a resonating body is a point or area of the resonant motion having the largest displacement, and an “electrode” is a structure through which an electrical or electromechanical effect is applied and/or sensed. In exemplary embodiments, various electrodes are used for driving resonators into their targeted mode shape at the designed frequency and/or sensing electrical or electromechanical effects through capacitive coupling (e.g., between a resonator mass and one or more adjacent structures), although it should be noted that other types of electrodes and couplings may be used (e.g., piezoelectric). Thus, in exemplary embodiments, electrodes may include a resonator mass and one or more structures for driving and/or sensing movement of the resonator mass.

FIG. 1 is a schematic diagram showing a top view of the resonator and support tethers for a single-axis resonating beam gyroscope 100, in accordance with certain exemplary embodiments. Among other things, the gyroscope 100 includes a single resonant beam 102 supported at its nodal points with respect to a predetermined resonant mode by four thin tethers 104 that allow the beam 102 to freely resonate both in-plane and out-of-plane. The tethers 104 are anchored to an underlying substrate and suspend the beam 102 above the substrate so that the beam 102 is free to resonate both in-plane and out-of-plane. For sensing rotations about the axis of sensitivity “a”, the gyroscope 100 may be configured such that beam 102 is driven to resonate out-of-plane by a set of drive electrodes (discussed below) and to deflect in-plane when the gyroscope is rotated about the axis of sensitivity a, where such deflections can be sensed by a set of sense electrodes (discussed below). Alternatively, the gyroscope 100 may be configured such that the beam 102 is driven to resonate in-plane and to deflect out-of-plane when the gyroscope is rotated about the axis of sensitivity “a”.

Importantly, the tethers 104 do not extend above or through the nodal points of the beam 102 (i.e., the top edge of the tethers is at or below the top edge of the beam), which, among other things, facilitates fabrication of the accelerometer compared to support structures that extend above and/or through the beam. The tethers 104 preferably are configured to maximize the Q (quality factor) of the resonating beam. Generally speaking, the tethers are as thin as possible (e.g., approximately 5 um in one exemplary embodiment) and the length of the tethers depends at least partially on the dimensions of the beam (e.g., approximately 30 um length in one exemplary embodiment having a beam that is 450 um in length and 50 um in height and width). The beam 102 and the tethers 104 may be formed from a unitary layer of material.

FIG. 2 is a schematic diagram showing conceptual components of the resonant beam 102, in accordance with exemplary embodiments of the invention. The beam's attachment to the tethers 104 on opposite sides of the beam 102 at locations 204 and 208 conceptually divides the resonant beam 102 into a central section 206 and two tail sections 202 and 210. The locations 204 and 208 are preferably at nodal points of the resonant beam with respect to both the drive mode shape and the sense mode shape of the resonant beam for the resonant mode in which the resonator 100 is configured to operate.

It is important to note that the tail sections of each beam 202 and 210 are very important for allowing the drive mode shape and the sense mode shape to be carefully configured. In this respect, the resonant beam 102 can be characterized by a node ratio, which is a parameter quantifying how long the tail portion is relative to the length of the beam. FIG. 3 is a schematic diagram depicting two parameters, Lnode and L, for determining the node ratio of a resonant beam 102, in accordance with one exemplary embodiment, where Lnode is the length of the portion of the resonant beam 102 from the center point to the nominal nodal point, L is the length of the resonant beam 102, and the node ratio=Lnode/L. In exemplary embodiments, the target node ratio can be approximated by node ratio=Lnode/L=X/(2*(X+1)), where X is the order of the flexural mode for the beam. For example, the Lnode/L target node ratio for a fundamental (first order) flexural mode is approximately ¼=0.25 (actually 0.275 in certain specific exemplary embodiments, the Lnode/L target node ratio for a second order flexural mode is approximately 2/6=0.33, etc. Of course, an alternative node ratio value could be based on the length of the tail portion (e.g., Ltail=L−Lnode, alternative node ratio=Ltail/L).

FIG. 4 schematically shows a first possible drive mode and corresponding sense mode for a single-axis gyroscope of the type shown in FIG. 1. In this exemplary embodiment, the beam 102 is driven in an out-of-plane fundamental (first harmonic) flexural resonant mode, as depicted in FIG. 4(A), and the beam 102 deflects in-plane in a fundamental (first harmonic) flexural mode due to Coriolis forces when the gyroscope is rotated about the axis of sensitivity a, as depicted in FIG. 4(B). The tethers 104 are located at the nodal points of the beam 102 with respect to the drive and sense modes.

Furthermore, typical embodiments of the gyroscope shown in FIG. 1 use electrostatic drive and sense transduction, as opposed to piezoelectric transduction.

FIG. 5 schematically shows a first possible arrangement of electrostatic drive and sense electrodes to support the drive and sense mode shapes shown in FIG. 4. Here, the beam 102 is flanked by two in-plane sense electrodes S102 a and S102 b, one on each side of the central section, while a single out-of-plane drive electrode D102 is underlying the beam 102. The electrodes are typically placed at the anti-nodes of the respective mode shapes. The drive electrode D102 is shown with broken lines to indicate that it is underlying the beam 102, e.g., on a substrate underlying the beam 102.

During operation of the gyroscope, an alternating drive signal is applied to the drive electrode D102 to cause the beam 102 to resonate out-of-plane. In this exemplary embodiment, a differential gyroscope output signal is produced from the sense electrodes S102 a and S102 b [i.e., Output=S102 a−S102 b]. Conceptually, when there is no rotation about the axis of sensitivity, the beam 102 will be equidistant from both sense electrodes and therefore the capacitances between each of the sense electrodes and the beam 102 will be the same and hence the differential gyroscope output will be zero. However, when there is rotation about the axis of sensitivity causing the beam to deflect in-plane, the beam 102 will move closer to one sense electrode while moving further from the other sense electrode and therefore the capacitances between each of the sense electrodes and the beam 102 will be different and hence the differential gyroscope output will be non-zero and in proportion to the rate of rotation.

FIG. 6 is a schematic diagram for an exemplary gyroscope drive and sense circuit 600 for the gyroscope arrangement shown in FIG. 5. Among other things, the gyroscope drive and sense circuit 600 includes a drive circuit 602 and a sense circuit 604. The drive circuit 602 provides alternating drive signals to the drive electrode 102 at a nominal drive frequency fo. The sense circuit 604 receives the differential sense signals from the opposing sense electrodes S104 a and S104 b and demodulates the signals at the drive frequency fo and combines the demodulated signals differentially to produce the output signal.

In the exemplary embodiment described above with reference to FIGS. 4 and 5, the gyroscope is configured to operate with an out-of-plane drive mode and an in-plane sense mode, although it should be noted that in various alternative embodiments, the gyroscope may be configured to operate with an in-plane drive mode and an out-of-plane sense mode. Thus, for example, the gyroscope 100 could be configured to operate with an out-of-plane drive mode depicted in FIG. 4(B) and an in-plane sense mode depicted in FIG. 4(A).

In the exemplary embodiment described above with reference to FIGS. 4 and 5, the gyroscope is configured to operate in its fundamental (first harmonic) mode, although it should be noted that in various alternative embodiments, the gyroscope may be configured to operate in a higher-order mode. When a higher-order mode is used, additional tethers may be placed at the additional nodal points.

It also should be noted that electrodes additionally or alternatively may be placed adjacent to the tail ends of the beams in-plane and/or out-of-plane, e.g., to drive, sense, and/or adjust motion of the beam 102. FIG. 7 schematically shows an alternate arrangement of electrodes including a set of optional electrodes placed adjacent to the tail ends of the beam for driving, sensing, or adjusting motion of the beam including both in-plane and out-of-plane electrodes. Here, compared to the arrangement of electrostatic drive and sense electrodes shown in FIG. 5, the beam 102 is flanked by additional in-plane sense electrodes S112 a and S112 b on opposite sides of the tail ends, and additional out-of-plane drive electrodes D102 b are underlying the tail ends. The additional drive electrodes D102 b are shown with broken lines to indicate that they are underlying the beam 102, e.g., on a substrate underlying the beam 102.

During operation of the gyroscope, alternating drive signals can be applied to the drive electrodes D102 a and D102 b to cause the beam 102 to resonate out-of-plane, with the drive electrodes D102 b driven in anti-phase to the drive electrode D102 a. In this exemplary embodiment, a differential gyroscope output signal is produced from the sense electrodes [i.e., Output=(S102 a+S112 a)−(S102 b+S112 b)]. It is important to note that the additional sense electrodes S112 a are on the opposite side of the beam 102 from sense electrode S102 a, while the additional sense electrodes S112 b are on the opposite side of the beam 102 from sense electrode S102 b.

When there is no rotation about the axis of sensitivity, the beam 102 is nominally equidistant from all of the sense electrodes and therefore the differential gyroscope output is nominally zero. However, when there is rotation about the axis of sensitivity causing the beam to deflect in-plane, the beam 102 will move closer to one set of sense electrodes (e.g., sense electrodes S102 a and S112 a) while moving further from the other set of sense electrodes (e.g., sense electrodes S102 b and S112 b) and therefore the differential gyroscope output will be non-zero and in proportion to the rate of rotation.

In various alternative embodiments, the additional electrodes can be used for other purposes. For example, rather than using the additional electrodes D102 b for driving resonance of the beam 102, these electrodes could be used to sense resonance of the beam 102, e.g., to provide a feedback signal for a PLL-based drive circuit. Similarly, rather than using the additional electrodes S112 a and S112 b for sensing in-plane deflections of the beam 102, these electrodes could be used to compensate for erroneous in-plane movements of the beam 102 such as from manufacturing or other imbalances.

It should be noted that, in various alternative embodiments, the gyroscope of the type described above may be operate inversely, i.e., driven in-plane with out-of-plane sensing.

One potential problem with operation of a resonating beam gyroscope of the type discussed above is that the driven motion of the beam 102 (e.g., out-of-plane in the exemplary embodiments described above) can include off-axis movements of the beam 102 (e.g., sense-axis movements) that can cause erroneous non-zero differential output signals (often referred to as quadrature error). Such off-axis movements (often referred to as modal coupling) can be caused from various sources typically associated with fabrication imperfections in typical MEMS fabrication processes, such as unequal spring constants of the tethers 104, differences in the dimensions of the drive electrodes, differences in the gaps between the drive electrodes and the beam 102, and imperfections in the dimensions of the beam 102 (e.g., side wall angle). The manifestation of this problem is as follows: typically in response to an in-plane force, only in-plane displacement is desired; when modal coupling occurs, an in-plane force causes both the desired in-plane displacement in addition to some amount of out-of-plane displacement dependent upon the degree of the fabrication imperfection causing it. The opposite is also true: a purely out-of-plane force will result in both out-of-plane and in-plane displacements. In both cases, the undesired displacement is proportional to both the desired displacement and the degree of imperfection.

One way to reduce or “trim” this error source is to apply an electrostatic force which applies a force in the opposite direction of the undesired displacement in such a manner to null the displacement in the undesired direction. This can be done by having split electrodes on top or bottom of the device at locations of maximum displacement of the mode shape. A differential DC voltage applied to each electrode (Vbias +/− Vtune) can provide the required opposite force to null the undesired displacement.

FIG. 8 is a schematic diagram showing an arrangement of split electrodes for reducing modal coupling, in accordance with one exemplary embodiment. Specifically, as shown in FIG. 8(A), in this example, the arrangement of split electrodes includes two opposing sets of trim electrodes 802 and 804 placed above the beam 102 (referred to as QTRM− and QTRM+), each having four trim electrodes, two placed along the center section on one side of the beam 102 and one placed along each tail section on the other side of the beam 102. Such placement of the trim electrodes is due to the resonant mode of the beam 102, i.e., when the center section of the beam 102 moves in one in-plane (y-axis) direction, the tail sections of the beam 102 move in the other direction. This arrangement of trim electrodes placed above the beam 102 generally would be appropriate for an out-of-plane drive mode (i.e., in the x-axis direction shown in FIG. 8) although such arrangement also can be used for an in-plane drive mode; similar trim electrodes placed on the side(s) of the beam 102 generally would be appropriate for an in-plane drive mode (i.e., in the y-axis direction shown in FIG. 8) although such arrangement also can be used for an out-of-plane drive mode.

FIG. 8(B) is a cross-sectional view of a beam 102 having a width “W” and pair of opposing trim electrodes 802 and 804, which are spaced from the beam 102 by a gap “g”. Here, the trim electrode 802 is a member of the QTRM− set of trim electrodes and receives a compensating voltage signal VQ− while the trim electrode 804 is a member of the QTRM+ set of trim electrodes and receives a compensating voltage signal VQ+. As shown in FIG. 8(B), the trim electrodes are “partial-overlap” electrodes, i.e., when the beam 102 is at its nominal (center) position along the y-axis, only part of the width of each trim electrode (represented by “y_(o)”) overlaps with the beam 102, and the amount of overlap is the same for the trim electrodes on both sides of the beam 102. When the beam 102 experiences displacement in-plane in the y-axis, the amount of overlap between each trim electrode and the beam 102 changes and hence any force produced on the beam 102 by a particular trim electrode varies in proportion to the amount of displacement. Thus, for example, with reference again to FIG. 8(B), if the beam 102 were to move toward the left in the y-axis direction, then the amount of overlap between the trim electrode 802 and the beam 102 would increase and correspondingly the amount of overlap between the trim electrode 804 and the beam 102 would decrease.

In the example shown in FIG. 8, the capacitance between each trim electrode and the beam 102 depends on the nominal gap “g” as well as the x-axis displacement (x_(disp)) of the beam 102 from its nominal x-axis position (i.e., capacitance increases as the beam moves toward the trim electrode and decreases as the beam moves away from the trim electrode). Also, because the trim electrodes are variable overlap electrodes, the capacitance between each trim electrode and the beam 102 also depends on the y-axis displacement “y” of the beam 102 relative to the nominal y-axis position “y_(o)” of the beam 102 (i.e., capacitance increases as the overlap increases and decreases as the overlap decreases). The differential capacitances dC− and dC+ per unit length of the beam 102 with respect to the QTRM− and QTRM+ electrodes, respectively, can be represented as follows:

${dC}_{-} = \frac{\varepsilon_{0}{{dl}\left( {y_{0} - y} \right)}}{g - x_{disp}}$ ${dC}_{+} = \frac{\varepsilon_{0}{{dl}\left( {y_{0} + y} \right)}}{g - x_{disp}}$

As discussed above, even when there is no rotation of the gyroscope, the driven motion of the beam 102 can cause erroneous sense-axis displacements of the beam 102 through modal coupling, and such erroneous sense-axis displacements of the beam 102 are generally proportional to the drive-axis displacement of the beam. Therefore, in order to compensate for such modal coupling, a quadrature cancellation circuit provides correcting voltage signals VQ− and VQ+ on the QTRM− and QTRM+ electrodes, respectively, to produce correcting forces in the sense-axis that substantially cancel the unwanted sense-axis displacements. Thus, the effective y-axis correcting force Fy in this example can be characterized by:

$F_{y} = {\frac{1}{2}\frac{\partial C}{\partial y}V^{2}}$

and consequently:

${dF}_{y} = {\frac{1}{2}\frac{\varepsilon_{0}{dl}}{g - x_{disp}}\left( {{VQ}_{+}^{2} - {VQ}_{-}^{2}} \right)}$

The effective force per trim electrode can be characterized by:

${\left. {dF}_{y.{QTRM}} \right.\sim\frac{1}{2}}\frac{\varepsilon_{0}{dl}}{g}\left( {1 + \frac{x_{disp}}{g}} \right)\left( {{VQ}_{+}^{2} - {VQ}_{-}^{2}} \right)$

For a single QTRM electrode pair, the part of the force that is linearly proportional to the x-axis drive force can be characterized by:

${\left. k_{{xy},{QTRM}} \right.\sim\frac{1}{2}}\frac{\varepsilon_{0}}{g^{2}}\left( {{VQ}_{+}^{2} - {VQ}_{-}^{2}} \right){\int{\varphi_{mode}{l}}}$

where the integral component of the equation allows for weighting by mode shape.

It should be noted that trim electrodes of the type shown in FIG. 8 can be placed above and/or below the beam. Electrodes can be placed above the beam, for example, by including the electrodes on a cap wafer that is attached to the device wafer containing the beam. Alternatively, the electrodes can be formed in situ with the beam and tethers, for example, by depositing and patterning additional material layers above device layer containing the beam and tethers. In addition to, or in lieu of, the variable-overlap electrodes placed above the beam, similar variable-overlap electrode can be placed below the beam, for example, supported directly or indirectly by a substrate underlying the beam. Thus, in certain embodiments, variable-overlap electrodes may be placed both above and below the beam. Similar trim electrodes additionally or alternatively may be placed at the side(s) of the beam 102.

It should be noted that embodiments of the present invention may use any of a variety of transduction methods for driving and/or sensing, including, but not limited to, electrostatic transduction or piezoelectric transduction.

The present invention may be embodied in other specific forms without departing from the true scope of the invention, and numerous variations and modifications will be apparent to those skilled in the art based on the teachings herein. Any references to the “invention” are intended to refer to exemplary embodiments of the invention and should not be construed to refer to all embodiments of the invention unless the context otherwise requires. The described embodiments are to be considered in all respects only as illustrative and not restrictive. 

What is claimed is:
 1. A resonating beam gyroscope comprising: a rectangular beam arranged along a longitudinal axis in a device plane, the rectangular beam having a plurality of nodal points on both sides of the beam in the device plane with respect to a fundamental or higher order flexural drive mode, said nodal points being remote from the ends of said beam such that said beam includes a tail portion at each end of said beam; a set of tethers in the device plane, each tether coupled to the beam at a distinct nodal point, wherein a top edge of each tether is at or below a top edge of the beam ; a set of drive electrodes configured for driving the beam to resonate in the drive mode; and a set of sense electrodes configured for sensing deflections of the beam caused by rotation of the gyroscope about the longitudinal axis.
 2. A gyroscope according to claim 1, wherein the drive mode is out-of-plane and the deflections are in-plane.
 3. A gyroscope according to claim 1, wherein the drive mode is in-plane and the deflections are out-of-plane.
 4. A gyroscope according to claim 1, wherein the relative length of the tail portions are characterized by a node ratio of the beam Lnode/L approximately equal to X/(2*(X+1)), where Lnode is the length from the center of the beam to a nodal point, L is the total length of the beam, and X is the order of the flexural drive mode for the beam.
 5. A gyroscope according to claim 1, wherein the tethers are configured to maximize a quality factor of the beam.
 6. A gyroscope according to claim 1, wherein the electrodes are electrostatically coupled with the beam.
 7. A gyroscope according to claim 1, further comprising: a first set of variable-overlap trim electrodes; and a second set of variable-overlap trim electrodes opposing the first set of variable-overlap trim electrodes, wherein the first and second sets of variable-overlap trim electrodes are configured to produce forces in the direction of such deflections to compensate for erroneous beam deflections in the direction of such deflections.
 8. A gyroscope according to claim 1, wherein the erroneous beam deflections are in proportion to resonation of the beam in the drive mode.
 9. A method of operating a resonating beam gyroscope, the method comprising: driving a rectangular beam to resonate in a fundamental or higher order flexural drive mode, the beam arranged along a longitudinal axis in a device plane, the beam supported by a set of tethers coupled to the beam at nodal points on both sides of the beam in the device plane with respect to the fundamental or higher order flexural drive mode, said nodal points being remote from the ends of said beam such that said beam includes a tail portion at each end of said beam, wherein a top edge of each tether is at or below a top edge of the beam; and sensing deflections of the beam caused by rotation of the gyroscope about the longitudinal axis.
 10. A method according to claim 9, wherein the drive mode is out-of-plane and the deflections are in-plane.
 11. A method according to claim 9, wherein the drive mode is in-plane and the deflections are out-of-plane.
 12. A method according to claim 9, wherein the relative length of the tail portions are characterized by a node ratio of the beam Lnode/L approximately equal to X/(2*(X+1)), where Lnode is the length from the center of the beam to a nodal point, L is the total length of the beam, and X is the order of the flexural drive mode for the beam.
 13. A method according to claim 9, further comprising: providing compensation signals to sets of variable-overlap trim electrodes to produce forces in the direction of such deflections to compensate for erroneous beam deflections in the direction of such deflections.
 14. A method according to claim 13, wherein providing compensation signals to sets of variable-overlap trim electrodes comprises: providing compensation signals to a first set of trim electrodes including at least one electrode placed along a center portion on one side of the beam and at least one electrode placed along each tail section on the other side of the beam; and providing compensation signals to a second set of trim electrodes opposing the first set of trim electrodes, wherein the trim electrodes variably overlap the beam with respect to a direction of such deflections and are configured to produce forces in the direction of such deflections to compensate for erroneous beam deflections in the direction of such deflections.
 15. A gyroscope comprising: a resonator means including a rectangular beam arranged along a longitudinal axis in a device plane and a set of tethers coupled to the beam at nodal points on both sides of the beam in the device plane with respect to a fundamental or higher order flexural drive mode, said nodal points being remote from the ends of said beam such that said beam includes a tail portion at each end of said beam, wherein a top edge of each tether is at or below a top edge of the beam; means for driving the beam to resonate in the drive mode; and means for sensing deflections of the beam caused by rotation of the gyroscope about the longitudinal axis.
 16. A gyroscope according to claim 15, wherein the drive mode is out-of-plane and the deflections are in-plane.
 17. A gyroscope according to claim 15, wherein the drive mode is in-plane and the deflections are out-of-plane.
 18. A gyroscope according to claim 15, wherein the relative length of the tail portions are characterized by a node ratio of the beam Lnode/L approximately equal to X/(2*(X+1)), where Lnode is the length from the center of the beam to a nodal point, L is the total length of the beam, and X is the order of the flexural drive mode for the beam.
 19. A gyroscope according to claim 15, wherein the tethers are configured to maximize a quality factor of the beam.
 20. A gyroscope according to claim 15, further comprising: means for providing compensation signals to sets of variable-overlap trim electrodes to produce forces in the direction of such deflections to compensate for erroneous beam deflections in the direction of such deflections. 